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**Abbas0000****Member**- Registered: 2017-03-18
- Posts: 27

Can somebody tell me that why there's so much fuss about prime numbers?

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**Alg Num Theory****Member**- Registered: 2017-11-24
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A prime number is a number (positive integer) that is divisible only by 1 and itself (excluding the number 1 itself). Examples of prime numbers are 2, 3, 5, 7, 11, 13, ….

IMHO the most important practical use of prime numbers in this digital age is their use in encrypting passwords. In the RSA cryptosystem, for example, there is asymmetry between the public encryption key and the private decryption key, based on the practical difficulty of factorizing the product of two large prime numbers. The method is as follows:

A user of RSA creates and then publishes a public key based on two large prime numbers, along with an auxiliary value. The prime numbers must be kept secret. Anyone can use the public key to encrypt a message, but with currently published methods, and if the public key is large enough, only someone with knowledge of the prime numbers can decode the message feasibly.

Prime numbers crop up in many areas of mathematics. The fundamental theorem of arithmetic states that every integer greater than 1 either is a prime or can be factorized uniquely (apart from the order of factors) into prime numbers. In abstract algebra, the integers are thought of as forming a subring of the field of rational numbers, and algebraic-number theory tries to generalize the unique-factorization property of the integers to algebraic integers, which are subrings of algebraic-number fields. The study of such UFDs (unique-factorization domains) was motivated by the search for a proof of Fermat’s last theorem. A complete proof proved elusive – it was not until the last decade of the 2oth century that Andrew Wiles succeeded where all others had failed by proving the Taniyama–Shimura conjecture for semistable elliptic curves – and the best that could be achieved before the 20th century was Kummer’s proof using the theory of ideals that the equation

has no nonzero integer solutions in *x*, *y*, *z* when *n* is an odd regular prime (or a multiple thereof).

*Last edited by Alg Num Theory (2017-11-24 12:00:42)*

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**Abbas0000****Member**- Registered: 2017-03-18
- Posts: 27

Thanks for your effort

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**Abbas0000****Member**- Registered: 2017-03-18
- Posts: 27

But a question. How do thay encrypt something with public key?

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